Development and verification of a simplified hp-version of the least-squares collocation method for irregular domains

L. S. Bryndin ; V. A. Belyaev ; V. P. Shapeev

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 16 (2023) no. 3, pp. 35-50

Résumé

A new high-precision hp-version of the least-squares collocation method (hp-LSCM) for the numerical solution of elliptic problems in irregular domains is proposed, implemented, and verified. We use boundary irregular cells (i-cells) cut off from the cells of a rectangular grid by a boundary domain and their external parts for writing the collocation and matching equations in constructing an approximate solution. A separate solution is not constructed in small and (or) elongated non-independent i-cells. The solution is continued from neighboring independent cells, in which the outer (and inner in a multiply-connected domain) part of the domain boundary contained in these non-independent i-cells is used to write the boundary conditions. This approach significantly simplifies the computer implementation of the developed hp-LSCM in comparison with the previous well-recommended version without losing its efficiency. We show reducing the overdetermination ratio of a system of linear algebraic equations in comparison with its values in the traditional versions of LSCM when solving a biharmonic equation. The results are compared with those of other papers with a demonstration of the advantages of the new technique. We present the results of bending calculations of annular plates of various thicknesses in the framework of the Kirchhoff–Love and Reissner–Mindlin theories using hp-LSCM without shear locking.

Détail

Keywords: least-squares collocation method, Kirchhoff–Love theory, Reissner–Mindlin theory, biharmonic equation, irregular domain.

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